The continuous and discrete spectrum of an asymptotically straight leaky wire
Seminar Room 1, Newton Institute
AbstractThe approach to quantum graphs developed by Exner and his co-workers is based on a Hamiltonian which contains a singular potential term with a delta-function support on (in two dimensions) a curve C. Here we give conditions on the potential and on the geometry of C under which the associated spectrum is either a semi-infinite interval or the whole real line. The geometry is expressed in terms of a new and simpler concept of asymptotic straightness which does not rely on an asymptotic estimate for the curvature, and which is only imposed on disjoint long sections of C. We also discuss the case where C is a star graph with N rays and the lower spectrum is discrete. We obtain an estimate for the lowest eigenvalue and we contribute to the conjecture that this eigenvalue is maximised for a given N when the star graph is symmetric. A number of open spectral problems related to this work are mentioned. (Joint work with Malcolm Brown and Ian Wood.)
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